In mathematics, certain functions or groups of functions gain prominence and have dedicated articles explaining them in detail. A significant branch, known as special functions, emerged from statistics and mathematical physics. Modern perspectives distinguish large function spaces—which are infinite-dimensional and contain many "anonymous" functions—from special functions characterized by properties like symmetry or their links to harmonic analysis and group representations. For more, see the List of types of functions.
Elementary functions
Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...)
Algebraic functions
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
- Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
- Constant function: polynomial of degree zero, graph is a horizontal straight line
- Linear function: First degree polynomial, graph is a straight line.
- Quadratic function: Second degree polynomial, graph is a parabola.
- Cubic function: Third degree polynomial.
- Quartic function: Fourth degree polynomial.
- Quintic function: Fifth degree polynomial.
- Rational functions: A ratio of two polynomials.
- nth root
- Square root: Yields a number whose square is the given one.
- Cube root: Yields a number whose cube is the given one.
Elementary transcendental functions
Transcendental functions are functions that are not algebraic.
- Exponential function: raises a fixed number to a variable power.
- Hyperbolic functions: formally similar to the trigonometric functions.
- Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular functions.
- Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
- Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
- Periodic functions
- Trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine, haversine, hacoversine, havercosine, hacovercosine, Inverse trigonometric functions etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
Special functions
Main article: Special functions
Piecewise special functions
- Indicator function: maps x to either 1 or 0, depending on whether or not x belongs to some subset.
- Step function: A finite linear combination of indicator functions of half-open intervals.
- Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta function.
- Sawtooth wave
- Square wave
- Triangle wave
- Rectangular function
- Floor function: Largest integer less than or equal to a given number.
- Ceiling function: Smallest integer larger than or equal to a given number.
- Sign function: Returns only the sign of a number, as +1, −1 or 0.
- Absolute value: distance to the origin (zero point)
Arithmetic functions
Main article: Arithmetic function
- Sigma function: Sums of powers of divisors of a given natural number.
- Euler's totient function: Number of numbers coprime to (and not bigger than) a given one.
- Prime-counting function: Number of primes less than or equal to a given number.
- Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers.
- Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.
- Prime omega functions
- Chebyshev functions
- Liouville function, λ(n) = (–1)Ω(n)
- Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p
- Carmichael function
Antiderivatives of elementary functions
- Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem.
- Exponential integral
- Trigonometric integral: Including Sine Integral and Cosine Integral
- Inverse tangent integral
- Error function: An integral important for normal random variables.
- Fresnel integral: related to the error function; used in optics.
- Dawson function: occurs in probability.
- Faddeeva function
Gamma and related functions
- Gamma function: A generalization of the factorial function.
- Barnes G-function
- Beta function: Corresponding binomial coefficient analogue.
- Digamma function, Polygamma function
- Incomplete beta function
- Incomplete gamma function
- K-function
- Multivariate gamma function: A generalization of the Gamma function useful in multivariate statistics.
- Student's t-distribution
- Pi function Π ( z ) = z Γ ( z ) = ( z ) ! {\displaystyle \Pi (z)=z\Gamma (z)=(z)!}
Elliptic and related functions
- Elliptic integrals: Arising from the path length of ellipses; important in many applications. Alternate notations include:
- Nome
- Quarter period
- Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena.
- Theta functions
- Neville theta functions
- Modular lambda function
- Closely related are the modular forms, which include
Bessel and related functions
- Airy function
- Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics.
- Bessel–Clifford function
- Kelvin functions
- Legendre function: From the theory of spherical harmonics.
- Scorer's function
- Sinc function
- Hermite polynomials
- Laguerre polynomials
- Chebyshev polynomials
- Synchrotron function
Riemann zeta and related functions
- Riemann zeta function: A special case of Dirichlet series.
- Riemann Xi function
- Dirichlet eta function: An allied function.
- Dirichlet beta function
- Dirichlet L-function
- Hurwitz zeta function
- Legendre chi function
- Lerch transcendent
- Polylogarithm and related functions:
- Incomplete polylogarithm
- Clausen function
- Complete Fermi–Dirac integral, an alternate form of the polylogarithm.
- Dilogarithm
- Incomplete Fermi–Dirac integral
- Kummer's function
- Riesz function
Hypergeometric and related functions
- Hypergeometric functions: Versatile family of power series.
- Confluent hypergeometric function
- Associated Legendre functions
- Meijer G-function
- Fox H-function
Iterated exponential and related functions
Other standard special functions
- Lambert W function: Inverse of f(w) = w exp(w).
- Lamé function
- Mathieu function
- Mittag-Leffler function
- Painlevé transcendents
- Parabolic cylinder function
- Arithmetic–geometric mean
Miscellaneous functions
- Ackermann function: in the theory of computation, a computable function that is not primitive recursive.
- Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution, but sometimes informally referred to as a function, particularly by physicists and engineers.
- Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
- Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
- Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
- Minkowski's question mark function: Derivatives vanish on the rationals.
- Weierstrass function: is an example of continuous function that is nowhere differentiable
See also
- List of types of functions
- Test functions for optimization
- List of mathematical abbreviations
- List of special functions and eponyms
External links
- Special functions : A programmable special functions calculator.
- Special functions at EqWorld: The World of Mathematical Equations.